The invention relates to the field of process measurement and control industry.
The process measurement and control industry employs process variable transmitters to remotely monitor process variables associated with fluids such as flurries, liquids, vapors, gases, chemicals, pulp, petroleum, pharmaceuticals, food and other processing plants. Examples of process variables include pressure, temperature, flow, level, turbidity, concentration, chemical composition, pH and other properties.
Complex mathematical computations are required to determine some process variables. For example, in order to determine flow by measuring differential pressure across an orifice plate, the computation requires the determination of physical properties of the fluid such as fluid density and the gas expansion factor. Calculation of these parameters involves extensive computation which reduces the update time of the transmitter, requires more complex processing equipment and requires increased power.
One technique to reduce the complexity of the calculations is to simply use fixed approximations for some of the parameters. For example, fixed values can be stored in memory rather than calculated using precise formulas. A more accurate technique is the use of straight polynomial curve fitting. In this technique, a process variable is estimated by using a less than complex polynomial rather than perform the precise computation. Such a technique is described in WIPO Publication No. WO 97/04288, filed Jun. 20, 1997 and entitled xe2x80x9cTRANSMITTER FOR PROVIDING A SIGNAL INDICATIVE OF FLOW THROUGH A DIFFERENTIAL PRODUCER USING A SIMPLIFIED PROCESSxe2x80x9d and U.S. Pat. No. 5,606,513, issued Feb. 25, 1997 and entitled xe2x80x9cTRANSMITTER HAVING INPUT FOR RECEIVING A PROCESS VARIABLE FROM A REMOTE SENSORxe2x80x9d. For example, the square root of the reciprocal of the compressibility (Z) of a fluid can be approximated using a xe2x80x9cstraightxe2x80x9d polynomial in the form of:                                           1            ⁢                          /                        ⁢            Z                          =                              ∑                          n              =              0                                      n              =              j                                ⁢                      xe2x80x83                    ⁢                                    ∑                              m                =                0                                            m                =                k                                      ⁢                          xe2x80x83                        ⁢                                          A                                  m                  ,                  n                                            ⁢                              xe2x80x83                            ⁢                              P                m                            ⁢                              xe2x80x83                            ⁢                              T                                  -                  n                                                                                        (        1        )            
where P is the absolute pressure of the fluid, T is the absolute temperature and the coefficients Am,n are the coefficients of the interpolating polynomial.
FIGS. 3A and 3B illustrate errors in prior art polynomial curve fitting techniques. FIG. 3A shows the curve fit error for ethylene as a function of pressure for 10 different temperature values. The highest powers of pressure P and inverse temperature (1/T) in the interpolating polynomial are eight and six, respectively. The temperatures and pressures for ethylene are not far from saturation pressures and temperatures. The minimum number (63) of pressure and temperature points was used to determine the 63 coefficients Am,n of the interpolation polynomial set forth in Equation 1, with j=6 and k=8. FIG. 3A illustrates the large error associated with straight polynomial curve fitting, particularly at the pressure extremes. FIG. 3B illustrates an even greater error when 32 bit floating point numbers (with 24 bit mantissa) are used. As illustrated in FIG. 3B, the results are essentially meaningless, with errors exceeding 105 percent. In the example of FIG. 3B, the coefficients were determined with 64 bit numbers and the calculation of the fitting polynomial was performed with 32 bit floating point numbers. Note that 32 bits is the typical number of bits used in floating point math performed in microprocessors. The use of a least squares fitting technique reduces the roughness produced by the loss of significant digits. However, the errors produced with such curve fitting are still relatively large in some situations.
This approximation technique is inaccurate, particularly when the highest power in the polynomial exceeds three or four and the polynomial is formed with multiple independent variables. This inaccuracy can cause inaccuracies in the measured process variable. Typically, the only solution has been to use the exact equations which require powerful computers and high power.
An interpolating polynomial is used to determine a process variable in a transmitter. The interpolating polynomial is xe2x80x9corthogonalxe2x80x9d and provides an accurate estimation of the process variable without a significant increase in power consumption. The transmitter includes a sensor configured to couple to a process and having a sensor output related to the process variable. The transmitter also includes a microprocessor coupled to the sensor output and having a process variable output. The process variable output is an orthogonal-polynomial function of the sensor output. A transmitter output is configured to provide an output related to the interpolated process variable.